Econometric Inference for Regularized Riemann-Liouville Fractional Brownian Motion Second-Order Filtering, HAC-GMM, and Weak Identification

We develop econometric inference for a regularized Riemann-Liouville fractional Brownian motion observed along one long path. Kernel regularization creates a multiscale variance crossover that identifies the Hurst parameter H and the regularization scale ε, except at the Brownian point H = 1/2, where ε is structurally unidentified. First-order quadratic moments cease to admit a finite root-n long-run variance at H ≥ 3/4. Second-order filtering restores short memory for every H ∈ (0, 1) while preserving the identifying crossover. We establish a functional central limit theorem, consistent HAC covariance estimation, and feasible minimum-distance/HAC-GMM inference. Near H = 1/2, the smallest eigenvalue of the Godambe information matrix is of order (H -1/2) 2 , yielding the mixed rate √ n |H -1/2| for ˆ︁ ε. Monte Carlo evidence documents the filter-order transition, HAC bandwidth sensitivity, and weak-identification distortions predicted by the theory.